Proof that the square root of 2 is irrational - Example One

Objective

To prove that the √2 is an irrational number.

Proof By Contradiction/Infinite Descent

Assume √2 is a rational number, represented by the irreducible fraction x/y. So we have √2 = x/y, which after squaring a multiplying by y², becomes x² = 2y².

This tells us that x is even, because x² is even and even*even = even and odd*odd = odd. So we can now use x=2a.

So now we can rewrite x² = 2y² as (2a)² = 2y², halving both sides, we get y²=2a².

This tells us that y² is even, which means y is even as well.

So, since y and x are both even, they are both divisble by 2, and thus x/y is not irredusible. This argument can then be repeated to infinity, so the assumption that √2 is a rational number is incorrect.

quod erat demonstrandum